# Parity flow as ${\mathbb Z}_2$-valued spectral flow

## TL;DR

This paper explores the topology of Fredholm operator paths on real Hilbert spaces, introducing a new approach linking parity to ${
m Z}_2$-valued spectral flow and applying it to topological insulators and PDE bifurcations.

## Contribution

It presents an alternative analytic method to define parity via ${
m Z}_2$-spectral flow, connecting it to chiral structures and providing concrete examples including topological insulators.

## Key findings

- Parity reduces to ${
m Z}_2$-spectral flow of chiral skew-adjoints
- Introduces ${
m Z}_2$-index for Fredholm pairs of chiral structures
- Applications to topological insulators and PDE bifurcation analysis

## Abstract

This note is about the topology of the path space of linear Fredholm operators on a real Hilbert space. Fitzpatrick and Pejsachowicz introduced the parity of such a path, based on the Leray-Schauder degree of a path of parametrices. Here an alternative analytic approach is presented which reduces the parity to the ${\mathbb Z}_2$-valued spectral flow of an associated path of chiral skew-adjoints. Furthermore the related notion of ${\mathbb Z}_2$-index of a Fredholm pair of chiral complex structures is introduced and connected to the parity of a suitable path. Several non-trivial examples are provided. One of them concerns topological insulators, another an application to the bifurcation of a non-linear partial differential equation.